Heterogeneous Stressed State

In our discussion, we will restrict omselves to selected characteristic problems that we will address at the basic qualitative and quantitative level. However, at the same time, we will make an attempt to reveal the physical natme of the model. We will use this simpMed approach in the discussion of the laws of material failme and the principals of mechanical testing. 

To begin, let us address a conunon and a rather characteristic example  

The nonhomogeneons natnre of this state is manifested in the dependence on the variable radius measured from the axis of the cylinder, p. This dependence is the same along the entire length of the rod. In approaching this problem, it is essential to select a region within which the stressed state is uniform. Such a region can be found at a fixed distance from the axis along the entire rod. Consequently, to select a unit volume. 

Physical-Chemical Mechanics of Disperse Systems and Materials 

At the same time, the displacement CC = /(p) = pt)), and also CC = Hy(p). We have therefore obtained a solution to the first problon, that is, the description of a uniform strain in a given elanentary volume and the distribution of the deformation in the sample, that is, y = y(p) = We have not yet introdnced material properties (elasticity, plasticity, etc.) and can t make any conclusions regarding the acting stresses. 

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So far, we have tacitly assumed that the stresses were applied externally. However, stresses which are induced by local changes in component concentrations and the corresponding changes in the lattice parameters during transport and reaction are equally important. These self-stresses can strongly influence the course of a solid state reaction. Similarly, coherent, semicoherent, and even incoherent interfaces during heterogeneous solid state reactions are sources of (local and nonlocal) stress. The... 

The type of nonuniform composition discussed in this chapter is recorded by Ozawa (1989) in naturally occurring crystals of spinel. The stress state at the time when the anomalous compositions developed is not known, but the crystals described are good examples of the type of heterogeneity predicted in this chapter. 

Endo,H., Fujimoto.T., Nagasawa,M. Normal stress and shear stress in a viscoelastic liquid under steady state flow Effect of molecular weight heterogeneity. J. Polymer Sci. Pt.A-2 9,345-362 (1971). 

Boundaries between solids transmit shear stress, particularly if they are coherent or semicoherent. Therefore, the strain energy density near boundaries changes over the course of solid state reactions. Misfit dislocation networks connected with moving boundaries also change with time. They alter the transport properties at and near the interface. Even if we neglect all this, boundaries between heterogeneous phases are sites of a discontinuous structural change, which may occur cooperatively or by individual thermally activated steps. 

A defining characteristic of a solid is the ability to resist shear. Therefore, stress is an additional feature which has to be taken into account when the physical chemistry of solids is at issue. Gibbs treated the thermodynamics of stressed solids a century ago in his classic work Equilibrium of Heterogeneous Substances under the title The Conditions of Internal and External Equilibrium for Solids in Contact with Fluids with Regard to all Possible States of Strain of the Solid . We have already mentioned in the introduction that stress is an unavoidable result of chemical processes in solids. Let us therefore briefly discuss the basic concepts of the thermodynamics of stressed solids. 

One further reason for the development of residual stresses should be mentioned. This is the heterogeneity of the final state of a material which may occur even if the initial reactive mixture was homogeneous. This phenomenon is related to the differing diffusion rates of the various components of the reactive mass during a chemical reaction. This localized distribution of concentrations can be frozen upon solidification of the material. 

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